The Second Derivative and Concavity

How the slope itself is changing.

12 problems · about 22 minutes · FUN-4.A

What this lesson teaches

The student uses the second derivative to determine concavity, inflection points, and the nature of critical points.

The second derivative reports how the slope is changing.
Positive means concave up, negative means concave down.
At a critical point, the sign of the second derivative names a maximum or a minimum.

Straightforward practice. Get the method working first.

5 problems

f′(4) = 0 and f″(4) = −3. Max or min? 1 max, 2 min.
Answer 1
Why A maximum.
What does an inflection point require?
Answer The second derivative must change sign.
Why A sign change, not merely a zero.
f″(3) = 4. Concave up or down? 1 up, 2 down.
Answer 1
Why Positive is up.
f′(5) = 0 and f″(5) = 2. Max or min? 1 max, 2 min.
Answer 2
Why Concave up gives a valley.
f(x) = x², so f″(x) = 2. Concave up or down everywhere? 1 up, 2 down.
Answer 1
Why 2 is positive.

The same ideas with more to keep track of.

3 problems

f(x) = x³ − 3x, so f″(x) = 6x. What is f″(−2)?
Answer -12
Why 6 × −2.
f′(1) = 0 and f″(1) = 0. Is the test conclusive? 1 for yes, 0 for no.
Answer 0
Why Zero says nothing.
How many inflection points does a cubic have?
Answer 1
Why Its second derivative is linear.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each sign of the second derivative by what it means.
Answer Concave up: f″ = 3, The slope is increasing · Concave down: f″ = −7, The slope is decreasing
Why The second derivative is the rate of change of the slope.
f(x) = x⁴, so f″(x) = 12x². What is f″(2)?
Answer 48
Why 12 × 4.
The Test: f′(2) = 0 and f″(2) = −5. Is x = 2 a maximum or minimum? 1 max, 2 min.
Answer 1
Why A maximum.
The Inflection: f″(x) = 6x. At which x is there an inflection point?
Answer 0
Why x = 0.